INSTANTANEOUS COMPANDING OF QUANTIZED SIGNALS 679 



quantity is a constant determined by the statistical properties of the 

 class of signals being studied. 



With the present choice of an exponential distribution of amplitudes 

 to represent speech, [see (25)], we have seen that .4 takes on the value 

 l/'\/2 = 0.707. It develops that A is not very sensitive to the choice of 

 P(e), as may be judged by the values \/2/ir = 0.798, and \/3/2 = 0.866 

 which would replace 0.707 if (25) were replaced by Gaussian and rec- 

 tangular distributions, respectively. The value .4 = 1/ V'2 will be used 

 in all numerical calculations; changes in the value of A to describe other 

 classes of signals (e.g., the aforementioned Gaussian or rectangular dis- 

 tributions) will change the plotted results by no more than a fraction of 

 a decibel. 



5. Degree of Compression (/x) 



From the foregoing it is clear that the essence of the compandor's 

 behavior is embodied in the one remaining variable which appears in (8) 

 and (17): the compression parameter ix. 



The significance of ^ has already received preliminary attention in 

 connection with Figs. 3 to 5. Fig. 6, where comparison of behavior at 

 constant A^ is facilitated by the choice of y/zND as ordinate, exhibits 

 the behavior of the ratio Z) as a function of C at constant ju. It will be 

 observed that the curves in Fig. 6 do not extend below their common 

 tangent which is labeled -v/sA^/^m-min- The significance of this lower 

 bound may be discussed in terms of Fig. 7 and the hypothetical ensemble 

 of compandors to which we now direct our attention. 



B. Optimum Compandor Ensemble 



Consider the artificial situation in which our communication system 

 includes an ensemble of instantaneous compandors, the members of 

 which correspond to different values of /x in (8). Since companding im- 

 provement varies with signal strength, we permit ourselves the luxury 

 of measuring the volume (i.e., C) of the input signal in order to assign 

 the optimum degree of companding compatible with (8), to each indi- 

 vidual signal. The compandor assigned to a signal is characterized by that 

 particular value of the compression parameter, ^ = iXc , which is required 

 to minimize D for a particular \'alue of C. This critical compression param- 

 eter may be calculated from the reriuirement that 



[dD/dlJL]A.C=const = (35) 



